Difficult math problem

Question

1) A radioactive Potassium K 42 reduces to it's half in 12.5 hours.
a) how much of 20g sample is remaining after 8 hours?
b) how long will it be until 8 G remains?

2) $28,000 is invested at an annual interest rate of 4.2% compound monthly. Let A(t) represent the balance after T years

A) find the formula for A(t)

B) find the balance after 7 years

c) after how many years will the account double?


4) A rectangular field along a river is to be fenced with 2000ft of fencing. Find the dimension of the field with maximum area

 

 

 

Answer 1

1)

Let x = initial amount of K 42.   .5x = x(.5)^12.5/λ where λ = half life.

 

.5 = .5^12.5/λ  1=12.5/λ  λ=12.5 hrs

 

y=20(.5)^8/12.5  y=12.83 g

 

8 = 20(.5)^t/12.5 >> .4 = .5^t/12.5 >> ln(.4) = (t/12.5)ln(.5) >>  12.5(ln(.4)/ln(.5) = t    t=16.52 hrs

 

2)

A(t) = 28,000(1 + .042/12)^12t

 

A(7)=28,000(1 + .042/12)⁸⁴  =$37,550.67

 

56000 = 28000(1 + .042/12)^12t  >>  2 = 1.0035^12t >> ln(2) = (12t)ln(1.0035)>> t=16.53 years

 

3)

let x= width   let y = length

 

2x + 2y =2000 >>>  y=1000 - x

 

x(1000 - x) is the area.   1000x - x²  -b/(2a)  is value of x where area is max.  -1000/(-2) = 500 ft  

x=500ft y=500ft

 

Answer 2

You should post each question separately, especially when they are so different.  

 

 

Basically, the half-life questions are of a form like these, which are all equivalent identities -  

 

A) Final Weight  = Original Weight * (1/2)^{(Time / Halflife)}

 

B) Original Weight = Final weight * 2^{(Time/Halflife)}

C) Original Weight / Final Weight = 2 ^{(Time/Halflife)}

 

D) Log₂ (Original Weight / Final Weight) = Time/Halflife

 

E) Halflife * Log₂ (Original Weight / Final Weight) = Time

 

So, plug your numbers for 1a into equation A above, and plug your numbers for 1b into equation E, and you have your answers.  (Assuming your calculator has log to the base 2).